categories: Re: Exact adjunctions

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Wed, 21 Nov 2001, Marco Grandis wrote:

> There is a nice lemma on adjoint functors, purely 2-categorical, and
> certainly known to various colleagues.
>
> As I need it in a paper on homotopy, I would like to know if it is
> *published with proof*, somewhere.
> I also like to "advertise" it here, because I think it deserves to be known
> more widely.
>
> LEMMA.
>
> If,in an adjunction, any one of the four natural transformations which
> appear in the triangle identities is invertible, so are the other three.
> [Proof below.]
>
> A few years ago I was considering such adjunctions, which I was calling
> "connections" because adjunctions between ordered sets ("covariant Galois
> connections") are always of this type.
> Renato Betti was also considering them, under the name of "exact
> adjunctions" (which I now prefer, also because "connection" has already too
> many meanings).
> I learnt from him that "one condition is sufficient".
>
I can't provide a reference for this result, but my ex-students will
testify that it has been an exercise on the problem sets for my first-year
graduate course in category theory for at least the last ten years. Also,
in my review of Francis Borceux' "Handbook of Categorical Algebra" for
the Bulletin of the London Math Soc., I referred to the result as a
"shibboleth" for testing whether someone is a genuine category theorist:
if he recognizes it as something he's always known, then he is a true
category-theorist, otherwise not. (This in the context that Borceux
appeared not to be aware of the result.)

Incidentally, "idempotent adjunction" seems to me a better name than
"exact adjunction".

Peter Johnstone